Primitive permutation group

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In mathematics, a permutation group G acting on a set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X. Otherwise, if G does preserve a nontrivial partition, G is called imprimitive.

This terminology has been introduced in his last letter by Évariste Galois who called (in French) equation primitive an equation whose Galois group is primitive.[1]

In the same letter he stated also the following theorem.

If G is a primitive solvable group acting on a finite set X, then the order of X is a power of a prime number p, X may be identified with an affine space over the finite field with p elements and G acts on X as a subgroup of the affine group.

An imprimitive permutation group is an example of an induced representation; examples include coset representations G/H in cases where H is not a maximal subgroup. When H is maximal, the coset representation is primitive.

If the set X is finite, its cardinality is called the "degree" of G. The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937:

Degree 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 OEIS
Number 1 2 2 5 4 7 7 11 9 8 6 9 4 6 22 10 4 8 4 A000019

Note the large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and alternating group, are subgroups of the affine group on the 4-dimensional space over the 2-element finite field.

While primitive permutation groups are transitive by definition, not all transitive permutation groups are primitive. The requirement that a primitive group be transitive is necessary only when X is a 2-element set; otherwise, the condition that G preserves no nontrivial partition implies that G is transitive.

Examples[edit]

\eta=\begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1 \end{pmatrix}.

Both S_3 and the group generated by \eta are primitive.

\sigma=\begin{pmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 4 & 1 \end{pmatrix}.

The group generated by \sigma is not primitive, since the partition (X_1, X_2) where X_1 = \{1,3\} and X_2 = \{2,4\} is preserved under \sigma, i.e. \sigma(X_1) = X_2 and \sigma(X_2)=X_1.

  • Every transitive group of prime degree is primitive
  • The symmetric group S_n acting on the set \{1,\ldots,n\} is primitive for every n and the alternating group A_n acting on the set \{1,\ldots,n\} is primitive for every n > 2.

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